Isoperimetric inequalities for the first Neumann eigenvalue in Gauss space

Chiacchio, F.; Di Blasio, G.

Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 2, pp. 199-216

We provide isoperimetric Szegö–Weinberger type inequalities for the first nontrivial Neumann eigenvalue ${\mu }_1\left(\Omega \right)$ in Gauss space, where Ω is a possibly unbounded domain of ${ℝ}^N$. Our main result consists in showing that among all sets Ω of ${ℝ}^N$ symmetric about the origin, having prescribed Gaussian measure, ${\mu }_1\left(\Omega \right)$ is maximum if and only if Ω is the Euclidean ball centered at the origin.

Classification : 35B45, 35P15, 35J70
Keywords: Neumann eigenvalues, Symmetrization, Isoperimetric estimates

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